Thursday, 22 September 2016

The BB84 protocol for quantum key distribution

The most successful application of quantum theory to cryptography is quantum key distribution (QKD). The goal of QKD is to generate an identical string of bits that is privately shared between two parties, which we shall call Alice and Bob.

The particular QKD scheme that we will describe was proposed by Charles Bennett and Gilles Brassard in 1984, and is often referred to as BB84.

The protocol has two main parts, a quantum and classical phase. In the quantum phase, Alice sends single photons to Bob over some public quantum channel. In the classical phase, Alice and Bob need to talk to each other over an authenticated classical channel, that is, it can be public but they need to verify that they are talking to the correct person.

Friday, 15 July 2016

The CHSH game and quantum entanglement

A simple setting for demonstrating the usefulness of entanglement involves a two-player game known as the CHSH game. The game is a variant of an experimental setup (by Clauser, Horne, Shimony and Holt) that is often used to illustrate Bell's theorem.

We shall call the two players 2 players Alice and Bob. We will also have Charlie as a referee that decides if Alice and Bob wins the game. They can decide on any strategy before the game commences but they cannot communicate with each other once the game starts.

To begin, Charlie picks two uniformly random bits $x$ and $y$, and gives $x$ to Alice and $y$ to Bob. Alice answers the referee with bit $a$, while Bob replies with bit $b$. After getting $a$ and $b$, Charlie checks whether

$a \oplus b = xy  \mod{2}$,

that is, that the XOR of the output bits $a$ and $b$ is equal to the AND of input bits $x$ and $y$. If so, then Alice and Bob win the game.

Sunday, 22 March 2015

The uncertainty principle and wave-particle duality are equivalent

In the 17th century, physicists debated over the true nature of light. When he observed that light is split into different colors by a prism of glass, Isaac Newton hypothesized that light is composed of particles he called corpuscles, which undergo refraction when they accelerate into a denser medium.

At around the same time, Christian Huygens proposed that light is made up of waves. In his treatise published in 1690, he described how light propagated by means of spherical waves, and explained how reflection and refraction occurs.

In 1704, Newton published Opticks, expounding on his corpuscular theory of light. The debate raged over whether light was a particle or a wave for almost a century, and was not settled until Thomas Young's interference experiments with double slits, which could only be explained if light was a wave.

The story did not end there. In 1901, Max Planck was able to explain the energy curve of blackbody radiation by supposing that light was emitted in small packets of energy. Planck thought of this light particles, or quanta, as a convenient mathematical device and did not believe them to be real. However, when the photoelectric effect was discovered in 1905, Albert Einstein showed that it could be explained in terms of wave packets of light we now call photons. In 1927, Louis de Broglie constructed a pilot wave theory that attempted to explain how particle and wave aspects of light can coexist.

Friday, 13 March 2015

The monogamy of entanglement

Quantum information theory has taught us that entanglement is a useful resource for communication and information processing. As with any other resource, we would like to describe the properties of entanglement quantitatively, to help us determine the various ways in which it can be manipulated. One particular question that might come to mind is this: to what extent can entanglement be shared among different objects?

The answer to this question leads us to an important fundamental property of entanglement called monogamy: if two objects are maximally entangled to each other, then neither object is entangled to a third one. More generally, it says that the stronger the entanglement between two objects is, the weaker their entanglements are with other objects.

Friday, 4 July 2014

An arbitrary quantum cannot be cloned

One of the early important results in the study of quantum information is the no-cloning theorem, which tells us that there is no quantum operation that allows us to create multiple copies of an arbitrary quantum state.

This property is very different from what we expect from classical information, which you may reproduce as many times as you wish. For example, you can send a PDF file by email to many recipients while keeping a copy to yourself. The important point is that whatever the contents may be, you can make a duplicate of it.

Now consider a cloning machine M for qubits that can produce identical copies of the states |u) and |d):

M |u) |0) = M |u) |u),
M |d) |0) = M |d) |d) ,

where |0) denotes any fixed initial state for M. This is necessary in pretty much the same way you would need a blank piece of paper before you can photocopy a printed document.

Friday, 13 June 2014

How to teleport a qubit

One of the fascinating things we can do with quantum entanglement is a scheme called quantum teleportation. In the original proposal by Charlie Bennett, Gilles Brassar, Claude Crepeau, Richard Jozsa, Asher Peres and Bill Wootters, it describes a way to transmit an arbitrary quantum state between two parties who may be far apart, using only a Bell state shared between the two parties, a few qubit operations that each party can perform independently, and two bits of information that can be communicated by one party to the other.

Suppose Alice and Bob are in separate locations but they share a pair of electrons that are in the entangled state

|E) = (|u,u) + |d,d)) / sqrt(2)

where as usual |u) denotes the state of an electron having its spin pointing in the up-direction, |d) denotes that with spin in the down-direction, and |u,u) refer to the state of the first and second electrons, respectively. Let's say that Bob has the first electron on his side and Alice has the second electron on her side. 

Alice also possesses a third electron in the state

|q) = a |u) + b |d)

and she wants Bob to obtain this state. If Alice does not know what the value of a and b precisely, she can not clone the state and send a copy to Bob. However, since Alice and Bob have shared entanglement, it is possible to transfer the state of this electron into Bob's electron using teleportation, which is shown in the figure below. 

Thursday, 13 March 2014

A quantum version of Zeno's paradox

The quantum Zeno effect describes the situation where an unstable particle, say a radioactive atom, won’t decay if it is observed continuously. More generally, it says that if you repeatedly interact with a quantum system through measurement then you can effectively freeze its quantum state.

The phenomenon is named after a Greek philosopher of ancient times, Zeno of Elea. Zeno is known best for a set of paradoxes (we know of 9 of them) that he posed as arguments  against Aristotle’s concept of motion. Here we are interested in the arrow or fletcher’s paradox.

If you observe an arrow flying through the air at some particular instant in time, then it would have a definite position, meaning it isn’t moving at that specific moment. However, you can think of the arrow’s motion as happening one moment at a time. This says that motion must be impossible since it is made up of this long sequence of motionless moments. 

Of course, as far as we can tell, the world is not static and objects in it are not forever motionless. What’s lacking with Zeno’s assertion is the mathematical notion of continuity. Motion is possible because time doesn’t flow like a series of separate frames in a film but more like the seamless current of a steady stream.